# Quantum Models of the Atom

When we introduced atoms earlier (Lecture 4) we left off with Rutherford's discovery that atoms must be mostly empty space, with a tiny, massive, positively charged nucleus in the center and electrons circulating around it in some fashion. A tremendous problem with this picture is that an electron going in a circle (or other non-linear path) will give off light (energy) of an energy depending on its velocity and the radius of curvature. Notice that as energy is released the electron will go to a smaller orbit, give off more light etc. Under the classical picture an atom should collapse with a flash of light nearly instantaneously. We're still here, so there's a problem!

Another great, and confusing, discovery of the nineteenth century was that atoms give off line spectra rather than continuous spectra. [Fig. 7.6, p 300]

Under classical pictures we would expect a continuous spectra. This is because electrons give off light when they are forced to change directions or slow down. Thus line spectra indicate that electrons exist in some kind of structured environment within the atom - not every speed or "orbit" can be available to them.

As you might imagine the simplest line spectra is owned by hydrogen. By the late nineteenth century folks had found that the line spectra of hydrogen could be described by a simple, empirical, mathematical relationship, known as the Rydberg equation:

where R = the Rydberg constant, NOT the Gas constant, and n is an integer with n2 > n1.

It turns out that hydrogen's spectra consists of a repeating pattern of lines. The first set, the Lyman series, occurs in the ultraviolet (UV) and is described by the Rydberg equation with n1 = 1, the second, the Balmer series, occurs in the visible and is described by the Rydberg equation with n1 = 2,

The third set, the Paschen series, occurs in the infrared (IR) and is described by the Rydberg equation with n1 = 3.

## Bohr Model of the Atom

In 1913 Bohr presented a very successful model for the structure of the hydrogen atom, which neatly explained the experimental observations noted before, including the line spectra and Rydberg equation. In order to account for the stability of the atom and the existence of line spectra he postulated that electrons in hydrogen have a minimum possible energy, the ground state, and that higher, excited energy states are quantized. The essence of Bohr's theory can be summarized in the equation for the energy of the electron:

E = -2.178 x -18J (Z2/n2)

where Z is the charge on the nucleus and n is an integer.

Bohr's equation can be used to derive the Rydberg equation by recalling that the energy of a photon emitted by an atom will be the difference between two electron energy levels:

Ephoton= hc/ = E2 - E1

Substituting Z = 1 and plugging in values gives the Rydberg equation exactly! (Of course one reason it works so well is that Bohr used the data from spectroscopy to find the value of his constant - he "fudged it.")

This is a wonderful result, and it rocked the scientific community. It said that at the microscopic level the world is discontinuous. The only trouble is, Bohr's model only works precisely for hydrogen! For most other atoms it is a dismal failure, though with a bit of tune-up it works reasonably well for the low energy spectra of the Alkali metals. (Note that like hydrogen these metals have only one electron that participates in chemistry. The Bohr model's partial success implies that this active electron "sees" the rest of the atom as a single charge at the nucleus - the active electron is somehow "outside" of the remaining electrons and thus behaving much like hydrogen's single electron.)

## Quantum (Wave) Mechanics

For some time the scientific community struggled with the failure of the Bohr model and the strangeness of the microscopic world. In the mid 1920's Werner Heisenberg and Erwin Schrödinger independently came up with models of the atom which accurately predicted the behavior of all atoms in principle.

A basic assumption of these treatments is that electrons behave like waves in some sense, and their locations in an atom are described by equations for waves. A second assumption is that of Heisenberg's Uncertainty Principle. This states that we cannot simultaneously know the position and momentum of a particle. Mathematically:

x = (mv) ≥ h/4π

Schrödinger assumed that the electron's behavior could be described by a three dimensional standing wave. He derived an equation which described the amplitude of this wave. The simplest solution for the Schrödinger Equation for the ground state (1s) energy of a hydrogen atom is:

1s= Ae-Br

where A & B are constants, e is the base of the natural logs, and r is the radial distance from the nucleus.

This equation has little real meaning. However, the square of the value of psi () tells the probability of finding an electron at any given location.

## A Quantum Picture of the Atom

We've taken a brief look at the physics underlying atomic structure, focusing on Schrödinger's Equation and the wave picture of electron distribution in atoms. Let's flesh this out a bit.

What we need to explain is the energy distribution of electrons in atoms and how this correlates with atomic properties. First recall the line spectrum of hydrogen and the Bohr model. We are going to keep the concepts of ground state and quantized energy levels from Bohr, after all they worked very well for Hydrogen. But we will need to build a new structure which will give these same predictions but with other factors which explain the details of hydrogen's spectra as well as other atoms. We'll again start by modelling hydrogen.

Electronic Energy Levels:

• We will designate the primary energy level, corresponding to the average radial distance of the electron from the nucleus as a shell, and give it the symbol n. The lowest possible energy level is then the ground state with n = 1.
• The value of n also gives the number of nodes in each of the orbitals in that shell, with each shell having one node at infinity, where:
• A node is a region of zero probability of finding an electron.
• Nodes can have two general geometries:
• radial (or spherical, since they describe a spherical shell at a specific radial distance from the nucleus), with each atom having at least one radial (spherical) node at infinity;
• angular (either planar, e.g. as in the planar p-node and diagonal d-nodes, or cone shaped, e.g. as in the cone-shaped nodes of the dz2 orbitals which results in the donut shaped orbitals).
• Shells with n > 1 may have subshells which are different geometrical patterns of electron distribution. Thus:
• The lowest energy pattern is spherical and given the designation s.
• The next lowest energy distribution is bi-lobed with a planar symmetry. It is given the designation p.
• The third lowest energy distribution has diagonal planes of symmetry and is designated d.
• The fourth lowest energy distribution is designated f. This is the highest subshell type occupied by ground state electrons in any atom, so we will not look any further (an infinite number of subshells exist in theory for excited states, but they are not important to our understanding).
• The average energies of the different subshells are the energy of the shell, thus when subshells are present the energy of the shell is split. For example, in the n=2 shell the 2s orbital becomes lower in energy than the shell, while the 2p orbitals become higher in energy.
• The regions of electron occupancy in subshells are called orbitals.
• For each shell there is one s orbital.
• For each shell with n = 2 or greater there is one s orbital and three p orbitals: px, py, and pz.
• For each shell with n = 3 or greater there is one s orbital, three p orbitals and five d orbitals: dxz, dyz, dxy, dx2- y2, and dz2

Atomic Orbitals Supplement More orbitals and some additional explanations.

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